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UT Arlington PHYS 3446 - Nuclear Models

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PHYS 3446 – Lecture #9Wednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt1PHYS 3446 – Lecture #9Wednesday, Sept. 24, 2008Dr. Andrew Brandt• Nuclear Models-Liquid Drop Model-Fermi Gas Model-Shell Model -Collective Model• Nuclear RadiationWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt2• Experiments showed that the nuclear forces have different characteristics than other forces• Quantifying the nuclear forces and understanding the structure of the nucleus were not straightforward• Several phenomenological models (not theories) developed that describe subsets of the experimental observations• Most of the models assume a central potential, like the Coulomb potentialNuclear ModelsWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt3• This was the earliest phenomenological model and had success in describing the binding energy of a nucleus• Nucleus is essentially spherical with radius proportional to A1/3. – Densities are independent of the number of nucleons• Led to a model that envisions the nucleus as an incompressible liquid droplet– In this model, nucleons are equivalent to molecules• Quantum properties of individual nucleons are ignoredNuclear Models: Liquid Droplet ModelWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt4• An early attempt to incorporate quantum effects• Assumes nucleus as a gas of free protons and neutrons confined to the nuclear volume– The nucleons occupy quantized (discrete) energy levels– Nucleons are moving inside a spherically symmetric well with the range determined by the radius of the nucleus– Depth of the well is adjusted to obtain correct binding energyNuclear Models: Fermi Gas ModelWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt5• Nucleons are Fermions (spin ½ particles) so– Obey Pauli exclusion principle– Any given energy level can be occupied by at most two identical nucleons – opposite spin projections• For greater stability, the energy levels fill up from the bottom to the Fermi level– Fermi level: Highest, fully occupied energy level (EF)• Binding energies are given as follows:– BE of the last nucleon= EF since no fermions above EFNuclear Models: Fermi Gas ModelWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt6• Experimental observations show BE is charge independent• If the well depth is the same for p and n, BE for the last nucleon would be charge dependent for heavy nuclei (Why?)– Since there are more neutrons than protons, neutrons would have higher EF•EFmust be the same for protons and neutrons. How do we make this happen?– Make protons move to a shallower potential wellNuclear Models: Fermi Gas Model• What happens if this weren’t the case?– Nucleus would be unstable– All neutrons at higher energy levels would undergo a β-decay and transition to lower proton levelsWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt7• Orbits and energy levels an electron can occupy are labeled by– Principle quantum number: n•ncan only be integer– For given n, energy degenerate orbital angular momentum: l• The values are given from 0 to n – 1 for each n– For any given orbital angular momentum, there are (2l+1) sub-states: ml•ml=-l, -l+1, …, 0, 1, …, l – l, l• Due to rotational symmetry of the Coulomb potential, all these sub-states are degenerate in energy– Since electrons are fermions w/ intrinsic spin angular momentum , • Each of the sub-states can be occupied by two electrons– So the total number of state is 2(2l+1)Atomic Shell Model Review2hWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt8• Nuclei are observed to have magic numbers just like inert atoms– Atoms: Z=2, 10, 18, 36, 54– Nuclei: N=2, 8, 20, 28, 50, 82, and 126 and Z=2, 8, 20, 28, 50, and 82 – Magic Nuclei: Nuclei with either N or Z a magic number Î Stable– Doubly magic nuclei: Nuclei with both N and Z magic numbers ÎParticularly stable• Could explain the stability of nucleus—but can we obtain these magic numbers with a simple model? Nuclear Models: Shell ModelWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt9• Exploit the success of atomic model– Uses orbital structure of nucleons– Nucleon energy levels are quantized– Limited number of nucleons in each level based on available spin and angular momentum configurations•For nthenergy level, l angular momentum (l<n), one expects a total of 2(2l+1) possible degenerate states for nucleonNuclear Models: Shell ModelWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt10• To solve equation of motion in quantum mechanics, Schrödinger equation, one must know the shape of the potential–– Details of nuclear potential not well known• A few models of potential attempted– Infinite square well: Each shell can contain up to 2(2l+1) nucleons• Can predict 2, 8 and 50 but no other magic numbers– Three dimensional harmonic oscillator• Can predict 2, 8 (but other predictions are wrong)Shell Model: Various Potential Shapes()()()2220mEVr rϕ⎛⎞∇+ − =⎜⎟⎝⎠rrh()Vr=2212mrϖWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt11Nuclear Models: Shell Model – Square well potential caseNMnl=n-1Ns=2(2l+1)NT21 0 2282 0,1 2+6820 3 0,1,2 2+6+10 1828 4 0,1,2,3 2+6+10+14 3250 5 0,1,2,3,4 2+6+10+14+185082 6 0,1,2,3,4,5 2+6+10+14+18+22 72Partial SuccessWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt12• Central potential could not reproduce all magic numbers• In 1940, Mayer and Jensen proposed a central potential + strong spin-orbit interaction w/ –f(r) is an arbitrary empirical function of radial coordinates and chosen to fit the data• The spin-orbit interaction with the properly chosen f(r), a finite square well can split• Reproduces all the desired magic numbersShell Model: Spin-Orbit PotentialTOTV =Spectroscopic notation: n L jOrbit numberOrbital angular momentumProjection of total momentum()Vr()frLS−⋅rrWednesday, Sept. 24, 2008 PHYS 3446, Fall 2008Andrew Brandt13• Spin-Parity of a large number of odd-A nuclei predicted well– Nucleons are Fermions so the obey Pauli exclusion principle – Î Fill up ground state energy levels in pairs– Ground state of all even-even nuclei have zero total angular momentum • The shell model cannot predict stable odd-odd nuclei spins– No prescription for how to combine the unpaired proton and neutron spinsPredictions of the Shell


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UT Arlington PHYS 3446 - Nuclear Models

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